Document Type : Research Paper
Author
Department of Mechanical Engineering, University of Qom, Qom, Iran
Abstract
Keywords
Main Subjects
Transient Solution of Heat and Moisture Transfer in a Functionally Graded Hollow Cylinder with Finite Length
Department of Mechanical Engineering, Faculty of Engineering, University of Qom, Qom, Iran
KEYWORDS 

ABSTRACT 
Transient hygrothermal; Coupled temperature and moisture; FGM; Hollow cylinder; DQM; 
In this article, temperature and moisture distributions in a finitelength hollow cylinder made of functionally graded material (FGM) under transient coupled hygrothermal condition was presented. The coupled equations of heat conduction and moisture diffusion were solved employing the Fourier series expansion method along the longitudinal direction, the differential quadrature method (DQM) along the radial direction, and the Newmark method for the time domain. Finally, the history of temperature and moisture potentials was obtained. The effect of coupled and uncoupled hygrothermal loading, grading index, and hygrothermal boundary conditions was illustrated in numerical examples. Results show that the difference between the coupled model and uncoupled model is more obvious in the moisture rather than the temperature. Also, the negative grading index increases the temperature and moisture. While the effect of the positive grading index is vice versa. Moreover, to have a more accurate analysis of the transient hygrothermal process, it is important to employ the coupled model. 
FGMs have been employed in industrial structures owing to their excellent thermomechanical properties. FGM refers to the materials whose properties vary gradually from one side to another side and are mostly used where the high thermal gradient exists. The idea of FGM is proper to various engineering areas such as nuclear energy, aerospace vehicle, chemical plant, and electronics [13]. In the last decade, many researchers tried to study the behavior of FGMs in various environmental conditions. Due to usages of industrial structures subjected to various environmental conditions and loads, disclosing the influence of humidity and thermal environment on their behavior is a new topic in the literature.
Using a finite elementfinite difference method, transient temperature distribution and thermal stresses in FGMs were presented by Wang et al. [4]. Jabbari et al. [5] studied the threedimensional thermomechanical stresses in FGM short hollow cylinders. Shao et al. [6] investigated thermoelastic analysis of FGM hollow cylinders under axisymmetric mechanical and transient thermal loads. Ootao and Tanigawa [7] analyzed a transient thermoelastic problem in an FGM hollow cylinder subjected to asymmetrical heating from its surfaces. Transient heat conduction in twodimensional FGM short hollow cylinders was analyzed by Asghari and Akhlaghi [8]. Peng and Li [9] investigated the transient response of temperature distribution and thermal stresses in an FGM hollow cylinder. Using a multilayered approach, the solution of temperature in FGM hollow cylinders under transient thermal boundary conditions was presented by presented Rahmati Nezhad et al. [10]. Using finite difference and Newmark methods, the twodimensional thermomechanical dynamic behavior of a long hollow FGM cylinder under asymmetrical thermomechanical loads was studied by Xie et al. [11]. Nejad and Kashkoli [12] presented timedependent thermomechanical creep analysis for a rotating hollow cylinder made of FGM. Hosseini and Dini [13] developed an analytical solution for the problem of stress analysis of rotating thickwalled FGM cylinders under the magnetic field and thermal loads at the plane strain state. Using DQM, Hosseini et al. [14] investigated strain gradient effects on the thermoelastic analysis of a microrotating FGM cylinder. The transient thermomechanical behavior of a short hollow cylinder made of twodimensional FGM was investigated by Najibi and Talebitooti [15]. Manthena and Kedar [16] considered the thermal stresses of a thickwalled hollow FGM cylinder with temperature dependent material properties. Using an analytical method, the problem of timedependent behavior of a functionally graded magnetoelectroelastic rotating hollow cylinder in a thermal environment was analyzed by Saadatfar [17]. Yarimpabuc obtained a closedform solution for transient thermal stress response of an FGM hollow cylinder subjected to temperature gradient under a periodic rotation [18]. A thermoelastic analysis for rotating FGM cylindrical shells with variable thickness under thermomechanical loading was presented by Jabbari and Zamani Nejad [19]. Based on the strain gradient theory, Dini et al. [20] presented a sizedependent analysis of the FGPM microrotating cylinder at the plane strain condition. Benslimane et al. [21] developed an analytical solution to evaluate stresses and deformations in a thickwalled cylindrical shell made of FGM.
Analysis of cylinders and cylindrical shells made of multilayered composites, FGMs, and smart materials in the hygrothermal environmental condition is of interest for many researchers. Saadatfar and AghaieKhafri [22] evaluated the hygrothermal stresses and deformations in a functionally graded magnetoelectroelastic (FGMEE) hollow sphere by means of an analytical method. Saadatfar and Aghaie Khafri [23] analyzed a smart cylindrical shell with FGM core and FGPM layers subjected to hygrothermal loading. They explained that the grading index has a significant effect of the actuation and sensing authority of FGPM layers subjected to multifield loading. Then, Saadatfar and AghaieKhafri analyzed the hygrothermoelectroelastic [24] response of a cylindrical shell made of FGM integrated with FGPM layers. Later, hygrothermomagnetoelectroelastic analysis of this problem was presented by saadatfar [25, 26]. Dini and Abolbashari [27] analyzed a hollow FGPM cylinder under a nonaxisymmetric hygrothermoelectroelastic loading. Saadatfar [28] investigated the history of displacement, stresses, electric potential, and magnetic potential of an FGMEE hollow cylinder under an axisymmetric hygrothermomagnetoelectromechanical loading for the plane strain condition. Saadatfar [29] presented an analytical solution for the problem of the timedependent response of a piezomagnetic rotating hollow cylinder under an axisymmetric multiphysical loading for the plane strain state.
Due to the complexity of coupled hygrothermal equations, the uncoupled hygrothermal equations have been supposed in most of the published articles. However, coupled hygrothermal equations for temperature and moisture should be considered to have an accurate evaluation for transient temperature distribution as well as moisture potential distribution. Using Laplace transformation and the finite difference method, Yang et al. [30] solved the transient coupled hygrothermal problem of a long cylinder. The distributions of the temperature, moisture, deformation, and stress of a rotating FGPM disk subjected to coupled hygrothermal loading were presented by Dai et al. [31]. Dai et al. [32] presented a mechanical model to study the coupled hygrothermal behavior of a variable thickness rotating porous disk made of functionally graded carbon nanotube reinforced composite. The transient coupled hygrothermal and mechanical behavior of a long cylinder under sudden hygrothermal loadings was studied by Peng et al. [33]. Peng et al. [34] analyzed the transient response of a porous long cylinder subjected to coupled hygrothermal condition.
Literature review reveals that there is not any reported work on transient responses of a shortlength FGM cylinder subjected to coupled hygrothermal condition. So, for the first time, the coupled equations of heat conduction and moisture diffusion were considered for shortlength FGM cylinder. The equations were solved using the Fourier series expansion method, DQM, and Newmark method. The distribution of temperature and moisture was obtained through time and radius. The effects of various parameters were investigated comprehensively in the numerical section.
Consider a short hollow cylinder with length L and inner and outer radii a and b, respectively. The cylinder is placed in a hygrothermal field and rotates around its axis with constant angular velocity. For the sake of simplifying, the material constants of the FGM are assumed to vary according to powerlaw function through the radial direction as:

where, Y and are material constants and corresponding values at the inside radius, respectively.
The coupled hygrothermal equations can be written as follows [35]:



Considering the cylindrical coordinate system, the Eqs. (2)(3) can be rewritten as:



Due to the axisymmetric condition, the terms include derivative respect to θ is vanished. Using Eq.(1), one has:



The initial conditions of the hygrothermal field are considered as:

The boundary conditions of the hygrothermal field can be considered as:

The solutions of the coupled hygrothermal problem satisfying the temperature and moisture concentration boundary conditions at the end faces can be considered as:

where p_{n}=np/L. Using Eq.(10), the Eqs. (6)(7) can be rewritten as:



Using the following formulation based on DQM, the n_{th}order derivative of the function f(x) at any sample point can be approximated as:

where, A_{ij}^{(n)} are the weighting coefficients used for the n_{th}order derivatives, and N is the number of grid points along the x direction. Employing the Lagrange interpolation polynomials as approximating functions, the following algebraic relations can be used to evaluate the first and the second order weighting coefficients:

For numerical computation, the Chebyshev–Gauss–Lobatto points with the following coordinates are used:

Using DQM, the equations of coupled thermal and moisture concentration fields can be written as:




Now the Newmark method is selected for discretizing the time domain. Divide the time domain by time step Dt. According to the Newmark method, the function U(r,t) and its derivatives with respect to time can be approximated by:

where a_{0}=1/4 and δ_{0}=1/2 are used. Now, Eqs.(16)(17) can be rewritten in the form as:

(19) 

(20) 
Now, a system of linear algebraic equations is obtained in each time step which can be solved using an iterative method. Domain and boundary degrees of freedom should be separated. They are indicated as (d) and (b) in vector forms, respectively. Accordingly, the matrix form of governing equations and boundary conditions can be expressed as:

(21) 
where:

(22) 
By eliminating the column vector Z_{b}; the matrix Eq. is reduced to the following system of algebraic equations:

(23) 
Employing direct or iterative methods, Eq.(23) can be solved as a system of algebraic equations. The direct methods may not be efficient for the large system of algebraic equations. In this case, employing iterative methods such as the GaussSeidel method is suggested.
To disclose the effects of different parameters on the transient behavior of the FGM hollow cylinder, some numerical examples are presented in this section. Material constants to be used for the FGM cylinder are in Table 1 [31, 33, 36]. The length, inside radius, and outside radius of the hollow cylinder are considered as L =1.5m a=0.2m and b=0.25m. To have accurate results, 27 nodes are considered through the thickness for DQM. The following nondimensional quantities are used in the results:

(24) 
To the best of author’s knowledge, there is no available paper in the literature for transient coupled hygrothermal analysis of FGM cylinders. So, the results were compared with reported results for a FGM cylinder subjected to transient thermal loading. The temperature distribution through the time was compared with Ref. [10]. In this case we have: r_{a}=0.5m, r_{b}=1m, L=1.5m, T_{a}=100(1e^{2t}), T_{b}=0. The used material constants can be found in the cited reference. The results are shown in Fig.1 As illustrated, the graphs show good agreement between the results.
Table 1. Material constants 

Parameter 
Value 
Parameter 
Value 
C_{p}(J/kg K) 
700 
x_{d} 
0.3 
k(W/mk) 
18.1 
γ 
0 
a (1/K) 
10.3´10^{6} 
ρ (kg/m^{3}) 
5300 
h_{LV} (kJ/kg) 
2500 
ζ_{i} (kg/ms°M) 
1.73´10^{8} 
ε (°M/K) 
2 
C_{m}(J/kg°M) 
0.01 




Fig. 1. Distribution of temperature along time 

(a) 

(b) 

Fig. 2. Distribution of (a) Tmeperature and 
(a) 
(b) 
(c) 
Fig. 3. Distribution of (a) temperature along radius, 

Fig. 4. Distribution of temperature, b=0.5 
The distribution of temperature and moisture concentration is studied now. In this case we have: T_{a}=350K, T_{b}=0K, M_{a}=0.015wt%, M_{b}=0.01wt%. To have a better understanding of changes through time, Fig. 2 shows the heat transfer and moisture diffusion along the radius and time (coupled model). It is observed that the boundary conditions, as well as initial conditions, are satisfied completely. For the next case, the difference between coupled model and the uncoupled model should be investigated. Fig. 3(a) shows the distribution of temperature along the radius. It shows that there is little difference between coupled hygrothermal with uncoupled model in temperature distribution. Fig. 3(b) shows the temperature distribution through time. Fig. 3(b) demonstrates that the FGM cylinder reaches the temperature balance earlier using the uncoupled model, and the temperature before balance increases with time. Using the coupled model, the time to reach the balance is longer and the temperature before the balance first increases and then decreases. Before reaching the balance, the temperature under the uncoupled model is lower than that by the coupled model. Fig. 3(c) depicts the distribution of moisture along the radius. It can be seen that the difference between a coupled model with an uncoupled model is more obvious in moisture rather than temperature. The difference becomes smaller by increasing the time.
For better understanding, Fig. 4 helps better understanding about difference between coupled and uncoupled model and the distribution of temperature along time and radius. Each line indicates the level of temperature that is denoted by a number above the line. Fig. 4 shows that the temperature transfers from inner radius to outer radius by serving the time. Each fixed point on the radius reaches the temperature levels earlier using the coupled model rather than the uncoupled model. Also, at a fixed time, the temperature transfers to a larger part of thickness using the coupled model rather than the uncoupled model.
Regarding the difference between the results under coupled and uncoupled model, for analyzing the transient conduction process it is vital to employ the coupled model. However, after the balance state, using the coupled model or uncoupled model has little effect.
The grading index is the next parameter that its effect was investigated. The effect of the grading index is demonstrated in Fig. 5 and Fig. 6. The boundary conditions are as previous. According to Fig. 5, the negative grading index increases the temperature and moisture at the same point, while the effect of the positive grading index is vice versa. Also, the negative grading index increases the balance temperature and the positive grading index reduces the balance temperature.
Fig. 6 shows the contour of heat transfer and moisture diffusion along time and radius for different grading indexes. It is observed that, similar to Fig. 5, using the negative grading index results in higher temperature transfer and moisture diffusion, and using the negative one has a reverse effect. According to Fig. 6(a), Each fixed point on the radius reaches the temperature levels earlier using a negative grading index. However, the effect of the positive grading index is vice versa. Also, at a fixed time, the temperature transfers to a larger part of thickness using the negative grading index. Whereas, a positive grading index has a reverse effect. According to Fig. 6(b), the effect of the grading index on the moisture transfer is similar to the temperature transfer.

(a) 

(b) 

(c) 
Fig. 5. Effect of inhomogeneity index on the distribution of 

(a) 

(b) 
Fig. 6. Effect of inhomogeneity index on the distribution of (a) Temperature (b) Moisture, based on coupled model 
The transient coupled heat transfer and moisture diffusion were considered for a short length FGM cylinder. The material constant of the cylinder was considered to vary according to a powerlaw function. The coupled equations of heat conduction and moisture diffusion were solved employing Fourier series expansion in the longitudinal direction, DQM in the radial direction, and the Newmark method in the time domain. According to the results, the FGM cylinder reaches the heat transfer balance earlier under the uncoupled model rather than the coupled model. Before reaching the balance, the temperature under the uncoupled model is lower than that by the coupled model. The difference between the coupled model with the uncoupled model is more obvious in moisture rather than temperature. Also, the negative grading index increases the temperature and moisture. While the effect of the positive grading index is vice versa. Moreover, for analyzing the transient hygrothermal process it is important to employ the coupled model.
Nomenclature
b 
Functionally graded index 
T 
Temperature change [K] 
M 
Moisture potential change [◦M] 
c_{p} 
Heat capacity [J/kg K] 
ρ 
Mass density [kg/m3] 
ζ 
Conduction coefficient of moisture [kg/ms◦M] 
x_{d} 
Ratio of vapor diffusion coefficient to coefficient of total moisture diffusion 
h_{LV} 
Heat of phase change [kJ/kg] 
L 
Length of cylinder 
ε 
Hygrothermal index 
c_{m} 
Moisture capacity [J/kg◦M] 
a, b 
Inner and outer raddi [m] 
k 
Thermal conductivity coefficients [W/m K] 
γ 
Heat of absorption or desorption [kJ/kg] 
A_{ij}^{(n)} 
Weighting coefficients used for the n_{th}order derivatives 
N 
Number of grid points 
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